ySEOtextz0. General Formulation, 1. Scalar Representation E0, E0={e|ÎE¸E,ÍÓ¸R(R3);(eÓ)(x)=Ó(E-1(x))}
To note the correspondence explicitly, let's define a mapping E0¸E0(E)
as follows. ÍE¸E;[E0(E)Ó](x)=Ó(E-1(x)) Then the scalar representation
of E+ is as follows. E+0={E0(E)|E¸E+} This is parametrized as follows.
E0(exp(-iÆL-iaP)=exp(-iÆL0-iaP0) where: (Lj0Ó)(x)=-iÃjklLxkÝlÓ(x),
(Pj0Ó)(x)=-iÝjÓ(x), {[Lj0,Lk0]=iÃjklLl0, [Pj0,Pk0]=0, [Lj0,Pk0]=iÃjklPl0,
2. General Case Es, Arbitrary linear representation of E can be constructed
as follows. First define generators asd it's domain of definition such
that the generators have the same commutation relations as E+0 and they
are linear. Second define the representation of E+ as a set of all operators
of the form: Es(exp(-iL-iaP))=exp(-iLs-iaPs) Finally find the representation
of the reflection -1¸E. Then we get: Es=E+s¾{eEs(-1)|e¸E+s}, 3. Symmetry
of Action